The structure of the walls of marginal stability for fixed charge \(\gamma \)  was analyzed in [1]}, [2]}, [3]}. The main result is that
for \(r>0\) , there is a finite number
of walls, forming two sequencesThis duplication is due to our definition
of DT invariants, which count semi-stable objects of charge \(\epsilon \gamma \) 
where \(\epsilon Z(\gamma )\in {H}_B\) .
of nested half-circles
on either side of a vertical wall at \(s=\frac{d}{r}\) .
For \((s,t)\)  outside the largest walls (sometimes called Gieseker walls) the index
\(\Omega _{(s,t)}(\gamma )\)  agrees with the Gieseker index \(\Omega _{\infty }(\gamma )\) , justifying the notation.
For \(r=0\)  and \(d\ne 0\) , there is a single sequence of nested walls, and
\(\Omega _{(s,t)}(\gamma )\)  similarly agrees with the Gieseker index \(\Omega _{\infty }(\gamma )\)  for outside the largest wall.
When \(\gamma \)  is a multiple of the Chern vector
associated to \(\mathcal {O}(m)\)  for some \(m\in {Z}\) , the chamber structure is actually trivial,
with \(\Omega _{(s,t)}(\gamma )=\delta _{r,1}\)  for any \(s<m\)  and \(t>0\) .
As each wall in the nested sequence is crossed,
the moduli space \(\mathcal {M}_\sigma (\gamma )\)  of \(\sigma \) -semistable objects undergoes a birational transformation contracting
a particular nef divisor associated to the wall (but keeping the dimension \(\dim _{C}\mathcal {M}_\sigma (\gamma )\) 
unchanged), until the innermost wall (sometimes known as collapsing wall) is crossed, after which \(\mathcal {M}_\sigma (\gamma )\)  becomes empty. In particular, the index vanishes at the point \((\mu \pm \sqrt{2\Delta },0)\)  or \((\textnormal {ch}_2/d,0)\) , respectively, where \(Z(\gamma )=0\) ,
which always lies inside the innermost wall.
In the rest of this section, we shall confirm this structure using the scattering diagram. Before
doing so, we make a couple of remarks.
